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Calculus and Beyond Homework Help
Field extensions of degree 10 over the rationals.
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[QUOTE="Theorem., post: 4547725, member: 257771"] [h2]Homework Statement [/h2] Show that if [itex] \alpha[/itex] is real and has degree 10 over [itex]\mathbb{Q}[/itex] then [itex]\mathbb{Q}[\alpha]=\mathbb{Q}[\alpha^3][/itex] [h2]Homework Equations[/h2] [h2]The Attempt at a Solution[/h2] It is clear that [itex]\mathbb{Q}[\alpha^3]\subset \mathbb{Q}[\alpha][/itex]. This gives us the sequence of fields [itex]\mathbb{Q}\subset \mathbb{Q}[\alpha^3]\subset \mathbb{Q}[\alpha][/itex]. Since these are finite extensions, we then have [itex][\mathbb{Q}[\alpha]:\mathbb{Q}]=10=[\mathbb{Q}[\alpha]:\mathbb{Q}[\alpha^3]][\mathbb{Q}[\alpha^3]:\mathbb{Q}][/itex]. Since [itex]x^3-\alpha^3\in \mathbb{Q}[\alpha^3][x][/itex] is of degree 3 and has [itex]\alpha[/itex] as a root, [itex][\mathbb{Q}[\alpha]:\mathbb{Q}[\alpha^3]]\leq 3[/itex]. Since it must also divide 10, it must be 1 or 2. The goal then is to show that it cannot be 2 (or then that [itex][\mathbb{Q}[\alpha^3]:\mathbb{Q}][/itex] cannot be 5.) I have tried going further on this point, but I think I'm just getting stubborn and missing something subtle. Any hints would be appreciated. -Theorem [/QUOTE]
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Calculus and Beyond Homework Help
Field extensions of degree 10 over the rationals.
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